MTH 532/632: Topology II (Differential Topology)

University of Miami, Spring 2023

Instructor: Christopher Scaduto
Email: c.scaduto @ math.miami.edu
Office: Ungar 525
Office hours: Tues/Thurs 10-11

Class Time and Location: 11:00-12:15 Tuesdays and Thursdays in Ungar 406.

Course syllabus can be found here.

Text: Differential Topology by Guillemin and Pollack

The symbol § used below means "Section".

Course Schedule

Date Lecture Content Reading Notes
1/17/23 Introduction. Smooth functions. Definition of manifolds. § 1.1
1/19/23 Examples of manifolds. Bump functions. § 1.1
1/24/23 Derivatives and tangent spaces. § 1.2
1/26/23 Hopf map, embedded 2-torus. Local structure of smooth maps: Rank Theorem. § 1.3, 1.4
1/31/23 Proof of Immersion Theorem. Preimage Theorem and examples. § 1.3, 1.4
2/2/23 Orthogonal group. Submanifolds, embeddings. Sard's Theorem. § 1.4, 1.7
2/7/23 Whitney embedding theorem. Fiber/vector bundles. § 1.8
2/9/23 Application to homotopy groups. Manifolds with boundary. § 1.6, 2.1
2/14/23 Brouwer Fixed Point Theorem. Morse Functions. § 2.2, 1.7
2/16/23 Abstract manifolds. Embedding abstract manifolds into Euclidean space.
2/21/23 Transversality. § 1.5, 2.3
2/23/23 Transversality continued. Proof of Transversality homotopy theorem. (Video lecture) § 2.3
2/28/23 Towards intersection theory mod 2. § 2.3, 2.4
3/2/23 EXAM 1
3/7/23 Intersection theory mod 2. Lusternik-Schnirelmann result. § 2.4
3/9/23 Degree theory mod 2. Jordan Brouwer Separation Theorem. § 2.4, 2.5
3/14/23 Spring break
3/16/23 Spring break
3/21/23 More mod 2 degrees. Jordan-Brouwer Separation. Whitney-Graustein. § 2.5
3/23/23 Orientations. Oriented manifolds. § 3.1, 3.2
3/28/23 Oriented intersection numbers. § 3.3
3/30/23 Euler characteristics and Lefschetz numbers. § 3.4
4/4/23 Vector fields. Poincare-Hop Theorem. Hopf degree. § 3.5, 3.6
4/6/23 Introduction to integration on manifolds. Exterior algebra. § 4.1, 4.2
4/11/23 Differential forms. Integration of forms on manifolds. § 4.3, 4.4
4/13/23
4/18/23
4/20/23
4/25/23
4/27/23



Homework Assignments



Assignment Due Date
Homework 1: Ch 1 § 1: # 2, 3, 12; Ch 1 § 2: # 2, 4, 8; Ch 1 § 4: #1 1/31/23
Homework 2: Ch 1 § 3: # 1, 2; Ch 1 § 4: 1, 2, 7; Ch 1 § 7: 1, 4 2/16/23
Homework 3: Ch 1 § 5: 2, 4, 7; Ch 1 § 6: 7; Ch 2 § 1: 4, 5 2/28/23
Homework 4: Ch 2 § 3: # 4; Ch 2 § 4: # 4, 5, 6, 7, 8, 11 3/23/23
Homework 5: Ch 4 § 2: # 5, 6; Ch 4 § 3: # 5, 7, 8, 9, 10, 12 4/18/23